Nonlinear evolution equations with variable domains in Hilbert spaces
Nobuyuki Kenmochi · Proceedings of the Japan Academy Series A Mathematical Sciences · 1977
Let H be a real Hilbert space and denote by (.,.) and 1]" the inner product and norm in H, respectively.Let Ct be a proper lower semi- continuous convex function on H and put Dt {v e H Ct(v) + oo} and D(t) {v e H; t(v) 4: i} or each t e [0, T], where 0 T + oo and is the subdifferential of Ct.In this paper we consider the evolution equationwhere u'(t)=(d/dt)u(t) and f is given in L(0, T; H).In recent years the evolution equation (E) with time-dependent domain D(3 t) has been studied by Attouch-Bnilan-Damlamian-Picard [1], Brzis [3], Moreau [7], Kenmochi [5] and Yamada [11].In the same direction we urther study the equation (E).For each 20 and t e [0, T], define (v) =in {I] v--z l]/ (2) + t(z) z e H}, v e H.According to [4; Chap.II], we see that (v)=(v-Jiv)/